Partial functional quantization and generalized bridges

Abstract

In this article, we develop a new approach to functional quantization, which consists in discretizing only a finite subset of the Karhunen–Loève coordinates of a continuous Gaussian semimartingale $X$.

Using filtration enlargement techniques, we prove that the conditional distribution of $X$ knowing its first Karhunen–Loève coordinates is a Gaussian semimartingale with respect to a larger filtration. This allows us to define the partial quantization of a solution of a stochastic differential equation with respect to $X$ by simply plugging the partial functional quantization of $X$ in the SDE.

Then we provide an upper bound of the $L^{p}$-partial quantization error for the solution of SDEs involving the $L^{p+\varepsilon}$-partial quantization error for $X$, for $\varepsilon>0$. The a.s. convergence is also investigated.

Incidentally, we show that the conditional distribution of a Gaussian semimartingale $X$, knowing that it stands in some given Voronoi cell of its functional quantization, is a (non-Gaussian) semimartingale. As a consequence, the functional stratification method developed in Corlay and Pagès [Functional quantization-based stratified sampling methods (2010) Preprint] amounted, in the case of solutions of SDEs, to using the Euler scheme of these SDEs in each Voronoi cell.